# Comparative Analysis of Starlight Occultation Data Processing

^{1}

^{2}

^{*}

## Abstract

**:**

^{15}. Furthermore, when applying the same method to the inversion of nitrogen trioxide and calculating the error, it was observed that the results of both methods were comparable at altitude of 30–60 km, with an error value ranging from 0 to 2%. However, at approximately 25 km, the inversion accuracy of the onion-peeling method surpassed that of the effective cross-sectional method. This research provides a theoretical foundation for further investigation of the stellar occultation inversion method and enhancing the accuracy of inversions.

## 1. Introduction

- Onion-peeling method: This method does not require any prior information and provides good vertical resolution. However, it is observed that the inversion of faint star data is noisy at low altitudes due to the strong influence of the observed star source, such as the apparent star. Therefore, it is preferable to use the observed data of bright stars for the inversion.
- Tikhonov-type regularization: Regularization, in linear algebra theory, refers to the fact that an ill-posed problem is usually defined by a set of linear algebraic equations, and that this set of equations usually stems from an ill-posed inverse problem with a large condition number. A large condition number means that rounding or other errors can seriously affect the outcome of the problem. As stellar occultation data inversion causes ill-posed problems, Tikhonov proposed using the ${\Vert Ax-b\Vert}^{2}+{\Vert \Gamma x\Vert}^{2}$A, $\Gamma $ method, called the Tikhonov matrix (Tikhonov matrix) [8]. Its advantage is its simplicity, but its disadvantage is that resolution is not taken into account and the optimal inversion wavelength depends on the spectral signal, as well as the tilt of the occultation.

## 2. Inversion Methodology

**s**is the light transmission path during an occultation event, ${\sigma}_{j}^{eff}$ is effective absorption cross-section, and ${N}_{j}$ is column density along the line of sight. By utilizing Equation (1) and T, the $\tau $ for different heights can be calculated. Consequently, with a known effective cross-section ${\sigma}_{j}^{eff}$, the column density ${N}_{j}$ can be calculated. The absorption cross-section at the tangent point can be read from the observation data. The above implements the first step of the algorithm. Assuming that the Earth has local spherical symmetry, Equation (2) [12] can be used to inversely obtain the vertical density profile of component j. This completes the second step of the algorithm.

**K**denotes the kernel matrix. Therefore, the core of the solution lies in the solution of the kernel matrix and line density matrix. The starlight occultation schematic is shown in Figure 1,

**AB**is the integral effective path, Z is the height of the tangent point of the occultation event, and 0, n, and j are the occultation event stratifications.

**H**denotes the height of each tangent point in Equation (8) [12].

**hj**=

**z**(

**j**− 1) −

**z**(

**j**). $\mathrm{Diag}[\frac{1}{{\mathrm{h}}_{\mathrm{i}}^{2}}]$ is the diagonal matrix, and the covariance matrix is generated.$\mathsf{\alpha}$ is related to the vertical resolution and is a regularisation parameter with a value of 10

^{15}[16,17]. At this point, the apparent magnitude of the corresponding target star is two (other conditions are described in Section 3).

**K**and

**N**are solved as follows and, from Equation (6), the density can be written as a function of the height. The derivation is as follows:

**S**is the effective path length of the occultation, which can be calculated based on the location of the tangent point and the coordinates of the receiving satellite position in the dataset. Z represents the height of the tangential point. For heights greater than 20 km, the light path deviation caused by refractive bending can be disregarded. This process is substituted into Equation (10) as follows:

**K**can be obtained. After the number density was obtained, the result was substituted into Equation (3) to obtain a new effective cross-section. The effective cross-section obtained from the calculation was substituted to obtain the vertical density profile, and the cycle was repeated once to obtain the desired result.

## 3. Analysis of Results

#### 3.1. Observation Data Sets

#### 3.2. Observation Data Processing

- (1)
- The apparent magnitude of the observed source was mag = 2.
- (2)
- Latitude selection was in the mid-latitude range of 30~60.
- (3)
- Occultation observation conditions: darklimb.
- (4)
- Temperature of the observed star source: greater than or equal to 10,000 K.

#### 3.3. Comparison of the Results of the Two Inversion Methods

#### 3.4. Inversion Results for Other Components

## 4. Conclusions

- The inversion results obtained through the effective cross-sectional method exhibited higher accuracy than the onion-peeling method when the inversion component was ozone, the observed source had an apparent magnitude of 2, the effective temperature exceeded 10,000 K, and the regularisation parameter was set to 10
^{15}. This is particularly evident in cases of low-altitude ozone inversion, for which an effective cross-sectional method is recommended. It is crucial to enhance the accuracy of the inversion within an altitude range of 20–45 km, which represents low-altitude regions in the adjacent space. - Under the same observational conditions used for the inversion of other components, such as nitric oxide, the precision obtained by the onion-peeling method at 25 km was approximately 4% higher than that obtained by the effective cross-sectional method. However, there was little difference between the two methods at other altitudes. This suggests that the parameters and conditions of the effective cross-sectional method are only suitable for ozone inversion, whereas the onion-peeling method is more versatile and applicable to a wider range of components.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Kyrölä, E.; Tamminen, J.; Leppelmeier, G.W.; Sofieva, V.; Hassinen, S.; Bertaux, J.L.; Hauchecorne, A.; Dalaudier, F.; Cot, C.; Korablev, O.; et al. GOMOS on Envisat: An overview. Adv. Space Res.
**2004**, 33, 1020–1028. [Google Scholar] [CrossRef] - Kyrölä, E.; Tamminen, J.; Leppelmeier, G.W.; Sofieva, V.; Hassinen, S.; Seppälä, A.; Verronen, P.T.; Bertaux, J.L.; Hauchecorne, A.; Dalaudier, F.; et al. Nighttime ozone profiles in the stratosphere and mesosphere bythe Global Ozone Monitoring by Occultation of Stars on Envisat. J. Geophys. Res.
**2006**, 111. [Google Scholar] [CrossRef] - Quémerais, E.; Bertaux, J.L.; Korablev, O.; Dimarellis, E.; Cot, C.; Sandel, B.R.; Fussen, D. Stellar Occultations observed by SPICAM on Mars Express. J. Geophys. Rev. 2006, in press.
- Lebonnois, S.; Quémerais, E.; Montmessin, F.; Lefèvre, F.; Perrier, S.; Bertaux, J.L.; Forget, F. Vertical distribution of ozone on Mars as measured by SPICAM/Mars-Express using stellar occultations. J. Geophys. Rev. 2006, in press.
- Gröller, H.; Montmessin, F.; Yelle, R.V.; Lefèvre, F.; Forget, F.; Schneider, N.M.; Koskinen, T.T.; Deighan, J.; Jain, S.K. MAVEN/IUVS stellar occultation measurements of Mars atmospheric structure and composition. J. Geophys. Res. Planets
**2018**, 123, 1449–1483. [Google Scholar] [CrossRef] - Jean-Loup Bertaux, D.; Nevejans, O.; Korablev, E.; Villard, E.; Quémerais, E.; Neefs, F.; Montmessin, F.; Leblanc, J.P.; Dubois, E.; Dimarellis, A.; et al. SPICAV on Venus Express: Three spectrometers to study the global structure and composition of the Venus atmosphere. Planet. Space Sci.
**2007**, 55, 1673–1700. [Google Scholar] [CrossRef] - Twomey, S. Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements; Elsevier Science: New York, NY, USA, 1977. [Google Scholar]
- Rodgers, C. Characterization and Error Analysis of Profifiles Retrieved from Remote Sounding Measurements. J. Geophys. Res.
**1990**, 95, 5587–5595. [Google Scholar] [CrossRef] - Sun, M.; Dong, X.; Zhu, Q.; Cheng, X.; Wang, H.; Wu, J. Comparison and Analysis of Stellar Occultation Simulation Results and SABER-Satellite-Measured Data in Near Space. Remote Sens.
**2022**, 14, 5065. [Google Scholar] [CrossRef] - Zhu, Q.; Sun, M.; Dong, X.; Zhu, P. Design and Simulation of Stellar Occultation Infrared Band Constellation. Remote Sens.
**2022**, 14, 3327. [Google Scholar] [CrossRef] - Sun, M.C.; Zhu, Q.L.; Dong, X.; Wu, J.J. Analysis of inversion error characteristics of stellar occultation simulation data. Earth Planet. Phys.
**2022**, 6, 61–69. [Google Scholar] [CrossRef] - Kyrölä, E.; Tamminen, J.; Sofieva, V.; Bertaux, J.L.; Hauchecorne, A.; Dalaudier, F.; Fussen, D.; Vanhellemont, F.; Fanton d’Andon, O.; Barrot, G.; et al. Retrieval of atmospheric parameters from GOMOS data. Atmos. Chem. Phys.
**2010**, 10, 11881–11903. [Google Scholar] [CrossRef] - Morozov, V.A. Regularization Methods for Ill-Posed Problems; CRC Press: Boca Raton, FL, USA, 1993. [Google Scholar]
- Tikhonov, A.; Arsenin, V. Solutions of Ill-Posed Problems; Wiley: New York, NY, USA, 1977. [Google Scholar]
- Tamminen, J.; Kyrölä, E. Bayesian solution for nonlinear and non-Gaussian inverse problems by Markov chain Monte Carlo method. J. Geophys. Res.
**2001**, 106, 14377–14390. [Google Scholar] [CrossRef] - Sofieva, V.F.; Tamminen, J.; Haario, H.; Kyrölä, E.; Lehtinen, M. Ozone profile smoothness as a priori information in the inversion of limb measurements. In Annales Geophysicae; Copernicus Publications: Göttingen, Germany, 2004; Volume 22, pp. 3411–3420. [Google Scholar]
- Honerkamp, J.; Weese, J. Tikhonovs regularization method for ill-posed problems: A comparison of different methods for the determination of the regularization parameter. Contin. Mech. Thermodyn.
**1990**, 2, 17–30. [Google Scholar] [CrossRef] - Bertaux, J.L. GOMOS Mission objectives. In Proceedings of the ESAMS99, European Symposium on Atmospheric Measurements from Space, Noordwijk, The Netherlands, 18–22 January 1999; WPP-161. ESA: Noordwijk, The Netherlands, 1999; pp. 79–87. [Google Scholar]
- Bertaux, J.L.; Pellinen, R.; Simon, P.; Chassefière, E.; Dalaudier, F.; Godin, S.; Goutail, F.; Hauchecorne, A.; Le Texier, H.; Mégie, G.; et al. GOMOS, proposal in response to ESA EPOP-1, A.O.1, January, 1988.
- Bertaux, J.L.; Mégie, G.; Widemann, T.; Chassefiere, E.; Pellinen, R.; Kyrölä, E.; Korpela, E.; Simon, P.C. Monitoring of ozone trend by stellar occultations: The Gomos instrument. Adv. Space Res.
**1991**, 11, 237–242. [Google Scholar] [CrossRef] - Bertaux, J.L.; Kyrölä, E.; Wehr, T. Stellar Occultation Technique for Atmospheric Ozone Monitoring: GOMOS on Envisat. Earth Obs. Q.
**2000**, 67, 17–20. [Google Scholar] - Bertaux, J.L.; Dalaudier, F.; Hauchecorne, A.; et al. Envisat: GOMOS-An instrument for global atmosphere ozone monitoring, edited by: Harris, RA[J]. ESA SP, 1244: 109.
- Bertaux, J.L.; Kyrölä, E.; Fussen, D.; Hauchecorne, A.; Dalaudier, F.; Sofieva, V.; Tamminen, J.; Vanhellemont, F.; Fanton d’Andon, O.; Barrot, G.; et al. Global ozone monitoring by occultation of stars: An overview of GOMOS measurements on ENVISAT. Atmos. Chem. Phys.
**2010**, 10, 12091–12148. [Google Scholar] [CrossRef] - Hansen, P.; Jacobsen, B.H.; Mosegaard, K. Methods and Applications of Inversion; Lecture Notes in Earth Science; Springer: Berlin/Heidelberg, Germany, 2000; Volume 92. [Google Scholar]
- Zhang, S.; Wu, X.; Su, M.; Hu, X. Inversion of ozone density by stripping onion of stellar occultation. Spectrosc. Spectr. Anal.
**2022**, 42, 203–209. (In Chinese) [Google Scholar]

**Figure 12.**Comparison of results from two inversion methods at heights above 50 km. It is shown that the inversion results of the two methods are consistent at altitudes above 50 km.

**Figure 13.**Comparison of results from two inversion methods at heights below 50 km. It is shown that there is a large difference between the inversion results of the two methods at altitudes below 50 km, especially at lower altitudes.

**Figure 14.**Error distribution of the two inversion methods at heights above 50 km. The figure shows that the errors of both inversion methods are around 1%, with the maximum not exceeding 1.2%, and there is no question of which inversion method is more accurate.

**Figure 15.**Error distribution of the two inversion methods for heights below 50 km. It can be seen that there is a difference in the accuracy of the two inversion methods, especially around 20–45 km, where the error obtained by utilizing the effective cross-sectional area method is smaller, with an overall error of around 1%.

**Figure 16.**Error distribution of two inversion methods for nitrogen trioxide. It can be seen that the inversion error obtained using the effective cross-sectional area method reaches 8%, while the inversion error obtained using the peeled onion method is better, with a maximum of 5%.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Sun, M.; Zhu, Q.; Dong, X.; Xu, B.; Wang, H.-G.; Cheng, X.
Comparative Analysis of Starlight Occultation Data Processing. *Atmosphere* **2023**, *14*, 1818.
https://doi.org/10.3390/atmos14121818

**AMA Style**

Sun M, Zhu Q, Dong X, Xu B, Wang H-G, Cheng X.
Comparative Analysis of Starlight Occultation Data Processing. *Atmosphere*. 2023; 14(12):1818.
https://doi.org/10.3390/atmos14121818

**Chicago/Turabian Style**

Sun, Mingchen, Qinglin Zhu, Xiang Dong, Bin Xu, Hong-Guang Wang, and Xuan Cheng.
2023. "Comparative Analysis of Starlight Occultation Data Processing" *Atmosphere* 14, no. 12: 1818.
https://doi.org/10.3390/atmos14121818